The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  X  X  X  X  X  X  X  1  1  1  X  1  1  1  1
 0 X^2  0 X^2  0 X^2 2X^2 2X^2 2X^2  0  0 X^2 X^2  0 X^2 2X^2 2X^2 2X^2  0  0 X^2 X^2  0 X^2 X^2 X^2 2X^2 2X^2  0 X^2 2X^2  0 2X^2 2X^2 2X^2  0  0  0 X^2 X^2
 0  0 X^2 2X^2 2X^2 X^2  0 X^2 2X^2  0 X^2 2X^2 X^2 2X^2  0  0 X^2 2X^2  0 X^2 2X^2 X^2 2X^2  0 2X^2 X^2  0 X^2 2X^2  0 2X^2 X^2  0 X^2 2X^2  0  0 X^2 2X^2 X^2

generates a code of length 40 over Z3[X]/(X^3) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+12x^78+210x^80+8x^81+4x^84+6x^89+2x^93

The gray image is a linear code over GF(3) with n=360, k=5 and d=234.
This code was found by Heurico 1.16 in 0.014 seconds.